4 One Redundant Internal Force Two Redundant Internal Force Compatibility Equations using Virtual Work Method General Procedure of Force Method Force Method for Structures with Temperature Changes Types of Temperature Changes Derivation of Force Method for Temperature Changes Force Method for Structures with Support Settlements Force Method for Structures with Elastic Supports and Joints Elastic Supports Elastic Joints Application of Force Method Complete Formulation of Force Method Estimation of Deflections Using Force Method Deflections Using Virtual Work Principle Deflections Using Alternate Virtual Work Principle Special Topics Displacement Method of Analysis: Slope-Deflection Method Structural Idealization Generalization of Frame Elements and Joints Slope-Deflection Equations for a General Frame Element Frame Element with Member-End Rotations and Moments Only Frame Element with Shear Deformations Only Frame Element with External Forces on the Frame Element Complete Formulation for a General Frame Element Representation in Terms of Stiffness Coefficients Slope-Deflection Equations for Special Frame Elements Frame with One End Pin-Supported Frame with Symmetric Deflections Frame with Antisymmetric Deflections Frame with Elastic Rotational Spring Application of Slope-Deflection Method for Continuous Beams Application of Slope Deflection Method for Frames: No Sidesway Frames with No Sidesway and Shear Deflection General Approach of Slope Deflection Method Special Cases Elastic Supports and Elastic Joints Application of Slope Deflection Method for Frames: With Sidesway Frames with Sidesway and Shear Deflections Examples of a Frame General Approach for Identifying Lateral Displacements Equations for Member-End Shear How to Write Equilibrium Equations for Shear General Approach of Slope-Deflection Method Symmetric Structures Definitions Symmetric Structures with Symmetric Loading: No Sidesway

22 Deformable Frames Under External Loads Work Done by External Forces External Forces Concentrated Distributed Displacements of Points where External Forces are Applied Concentrated Distributed Work Done by External Forces Concentrated Distributed >Integration Work Done by External Forces for Linear Elastic Frames Triangular Relations Work Done by Internal Forces for Frames: Strain Energy Internal forces Moment Shear Axial Internal Strains Bending Shear Axial Internal Force/Internal Strain Relations Work Done by Internal Forces Work Done by Internal Forces for Linear Elastic Frames Traingular Relations Relation between External Events and Internal Events for Elastic Frames Triangular Relation Theory of Conservation of Energy Case for One Set of External Loads Work Done by External Forces Work Done by Internal Forces Work Done by External Forces is Equal to Work Done by Internal Forces Linear Elastic Structures: Triangular Equality Case for Two Sets of External Loads Application of First Set Application of Second Set 21

23 9.5 Displacements by Virtual Work Virtual Work Theory Goal: Finding Displacements Two Sets of Loading First Set is a Unit Load Second Set is the Original Loading Application of Theory of Conservation of Energy Virtual Work Principle Sign Convention Procedure for Estimation of Displacements Temperature Changes Support Settlements 22

27 Chapter 11 Force Method of Analysis 11.1 Redundant Reactions One Redundant Reaction Primary Stable Determinate Structure with Original Loading Redundant Reaction (One Unknown) and Its Application to the Primary Structure Deflection of Indeterminate Structure at the Application Point of Reduntant Reaction is Zero Deriving Compatibility Equation by Manual Superposition of Deflections Coefficients of Equation using Virtual Work Method Superposition of Internal Forces and Reactions Sign Convention Two Redundant Reactions Primary Stable Determinate Structure with Original Loading Redundant Reactions (Two Unknowns) and Their Application to the Primary Structure Deflections of Indeterminate Structure at the Application Points of Reduntant Reactions are Zero Deriving Compatibility Equation by Manual Superposition of Deflections Coefficients of Equation using Virtual Work Method Superposition of Internal Forces and Reactions Sign Convention 26

28 11.2 Redundant Internal Forces Review of Internal Releases: Hinges Review of Relative Rotation Continuous Joint Joint with Hinge: Four Possible Cases Sign Convention One Redundant Internal Force Primary Stable Determinate Structure with Original Loading Redundant Internal Force (One Unknown) and Its Application to the Primary Structure. Definition of a New Sign Convention Deflection of Indeterminate Structure at the Application Point of Reduntant Reaction is Zero Deriving Compatibility Equation by Manual Superposition of Deflections. Definition of a New Sign Convention Coefficients of Equation using Virtual Work Method Superposition of Internal Forces and Reactions Two Redundant Internal Force Primary Stable Determinate Structure with Original Loading Redundant Internal Forces (Two Unknowns) and Their Application to the Primary Structure Deflections of Indeterminate Structure at the Application Points of Reduntant Reactions are Zero Deriving Compatibility Equation by Manual Superposition of Deflections Coefficients of Equation using Virtual Work Method Superposition of Internal Forces and Reactions 11.3 Compatibility Equations using Virtual Work Method Derivation of Compatibility Equations by Manual Superposition (Review) Derivation of Compatibility Equations by Virtual Work Method Identify Redundant Forces (Reactions and Internal Forces) 27

29 Identify Deflections of Indeterminate Structure at Redundant Force Locations as Zero Apply One Redundant Force to the Primary Structure as External Loading Define Virtual Displacements and Deformations as the Displacements and Deformations of Indeterminate Structure (Internal Forces under These Deformations are Designated) Apply Virtual Displacements and Deformations to the Primary Structure on Top of the Redundant Force Two Quantities of Work are Estimated Work Done by Internal Forces Due to Virtual Deformations Work Done by Redundant Forces Over Virtual Deflections at the Redundant Force Location. This is Zero if There is No Support Settlement. Two Quantities of Work are Equal to Each Other Due to Virtual Work Principle This Procedure Gives Closed-Form of Compatibility Equations Derivation of Equations of Superposition of Internal Forces Insertion of Equations of Superposition of Internal Forces into Closed-Form of Compatibility Equations Gives Open-Form of Compatibility Equations. Coefficients of Open-Form of Compatibility Equations are Displacements of Primary Stable Determinate Structure Under Unit Loadings that are Applied to Redundant Force Locations Maxwell s Reciprocal Theory and Betti s Law 11.4 General Procedure of Force Method Selection of Primary Stable Determinate Structure and Redundant Forces (Reactions and Internal Forces) Stating Primary Structure with Original Loading and Redundant Loading Primary Stable Determinate Structure with Original Loading. Internal Forces are Designated as: M 0, N 0, T 0 Primary Stable Determinate Structure with Redundant Loading (Generally Unit Load) for the i th Redundant Force. Internal Forces are Designated as: M i, N i, T i Deflections of Primary Structures. These Deflections are Coefficients of Open-Form of Compatibility Equations. Open-Form of Compatibility Equations are Derived. Equations are Solved for Unknown Redundant Forces Reactions and Internal Forces of the Indeterminate Structure are Obtained by Superposition of Primary Structure with Original Loading and Redundant Force Loadings. Internal Forces are Designated as: M, N, T 28

30 11.5 Force Method for Structures with Temperature Changes Types of Temperature Changes Constant Gradual Derivation of Force Method for Temperature Changes 11.6 Force Method for Structures with Support Settlements Case where support settlement occurs at a location of a Redundant Reaction Case where support settlement does not occur at a location of a Redundant Reaction General Approach 11.7 Force Method for Structures with Elastic Supports and Joints Elastic Supports Rotational Translational Estimation of Elastic Support Properties for Single Footings Elastic Joints Rotational Application of Force Method 11.8 Complete Formulation of Force Method 11.9 Estimation of Deflections Using Force Method Deflections Using Virtual Work Principle Deflections Using Alternate Virtual Work Principle Special Topics 29

34 Elastic Supports and Elastic Joints Slope-Deflection Equations for Elastic Supports and Elastic Joints Slope-Deflection Equations for a Frame Element with a Elastic Rotational Spring (Review) 12.7 Application of Slope Deflection Method for Frames: With Sidesway Frames with Sidesway and Shear Deflections Concept of Lateral Displacements and Story Drifts Dependent and Independent Drifts Examples of a Frame Frame with One Vertical Column and One Horizontal Beam Frame with One Inclined Columns and One Horizontal Beam Frame with Two Vertical Column and One Horizontal Beam Frame with One Inclined Column, One Vertical Column and One Horizontal Beam Frame with Two Inclined Column and One Horizontal Beam Discussion of Dependent and Independent Lateral Displacements Relation between Frame-End Displacements New Equilibrium Equations using Shear Forces General Approach for Identifying Lateral Displacements Number of Independent Lateral Displacements Finding the Relation between Lateral Displacements from Projection Equations Equations for Member-End Shear Two Methods Based on Equilibrium of Frame Element Forces Based on Stiffness Coefficients Fixed-End Shears 33

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